Question

1. in the accompanying diagram, line m is parallel to line n and they are cut by transversal t. if the acute angle shown is ( 15^{circ} ), which of the following statements is true? (1) ( m angle x = 15^{circ} ) and is an obtuse angle (2) ( m angle x = 15^{circ} ) and is an acute angle (3) ( m angle x = 165^{circ} ) and is an obtuse angle (4) ( m angle x = 165^{circ} ) and is an acute angle

Explanation:

Step1: Recall the property of parallel lines and transversal

When two parallel lines \(m\) and \(n\) are cut by a transversal \(t\), consecutive interior angles are supplementary. That is, if one angle is \(\alpha\) and the other is \(\beta\), then \(\alpha+\beta = 180^{\circ}\). Here, \(x\) and the \(15^{\circ}\) angle are consecutive interior angles.

Step2: Calculate the measure of \(\angle x\)

We know that \(m\angle x+15^{\circ}=180^{\circ}\) (by the consecutive - interior angles theorem for parallel lines \(m\parallel n\) and transversal \(t\)). Solving for \(m\angle x\), we get \(m\angle x=180^{\circ}- 15^{\circ}=165^{\circ}\).

Step3: Classify the angle

An acute angle \(\theta\) satisfies \(0^{\circ}<\theta < 90^{\circ}\), and an obtuse angle \(\varphi\) satisfies \(90^{\circ}<\varphi<180^{\circ}\). Since \(90^{\circ}<165^{\circ}<180^{\circ}\), \(\angle x\) is an obtuse angle.

Answer:

(3) \(m\angle x = 165^{\circ}\) and is an obtuse angle